Finite Element Micromechanical Analysis of Composite Materials

Summary

This article describes the finite element micromechanics analysis method applied to investigate the mechanical behaviour of composite materials. The study examines a composite system made from thermoset prepreg with fibres reinforced unidirectionally, [0], and cross-ply, [0/90]s. Three-Dimensional Representative Volume Elements (RVEs) are created using commercial finite element software according to the volumetric fiber ratio of the composite material. These models can calculate the elastic properties of composite materials with high accuracy. Additionally, stress distributions within the fibre and matrix can be examined within these models and can provide insight into where and how progressive damage will occur.

1. Introduction

The Finite Element Method is a numerical method widely used for providing predictions about the mechanical behaviour of materials. Because it provides insight into the mechanical behaviour of the materials being studied, it reduces experimental costs and is therefore a highly valuable method. The Finite Element Micromechanics Analysis Method is an important numerical method frequently used over the past twenty years for its ability to simulate the mechanical behaviour of materials at micro-scales. A considerable number of studies employing this method have been published. Although polymer composites are generally analyzed using this method, all other composite materials can also be numerically analyzed using this method. The mechanical behaviour of any type of material under different types of loading can be examined at the micro-scale. In the Finite Element Micromechanics Analysis Method, simple geometric models in which fibres and matrix are arranged together are created according to the volumetric fibre ratio of the material. Generally, fibres are placed within the model according to two geometric arrangements; these are the square and hexagonal regular models shown in Figure 1. For these geometries, two different material types must be defined for fibre and matrix in the program interface, and therefore the properties of the components must be known. It should be remembered that carbon and glass fibres are materials with orthotropic or transversely isotropic properties, while polymer matrices are materials with isotropic properties.

Figure 1. RVEs arranged in square and hexagonal patterns.

In this article, the mechanical behaviour of a carbon fibre reinforced thermoset polymer composite is investigated using RVEs created according to square and hexagonal arrangements with the finite element micromechanics analysis method. Model creation and analysis are carried out using ABAQUS®, a commercial finite element programme [1]. First, the finite element micromechanics analysis method applied in ABAQUS is described. Subsequently, the elasticity coefficients of the composite material are calculated using RVEs. Finally, the calculated results are compared with available data in the literature. As a result of this study, it is shown that the finite element micromechanics analysis method provides good agreement with experimental results. This agreement proves that the method makes effective predictions of composite material behaviour with simple models and low computation times.

2. Micromechanics Model 2.1. Representative Volume Elements and Boundary Conditions

The meshed RVEs are shown in Figure 2. These elements can be reduced to simpler dimensions shown by black lines due to symmetry properties. To ensure periodicity and symmetry in the model and results, "periodic boundary conditions" must be used [2, 3]. The equations defining periodic boundary conditions for node points in the model are shown in equations numbered 1-3. Applying these boundary conditions in the ABAQUS interface using "Symmetric Boundary Conditions" and "Equation Type Constraints" is quite straightforward. The ways in which periodic boundary conditions are defined in the ABAQUS interface are shown in Tables 1 and 2. The capital letters on the left of the tables (X,Y,Z) show the normal directions of the representative volume element planes, while the lowercase letters at the top (x,y,z) show the directions of the node elements in those planes. The material considered in this study is a thermoset prepreg composite system with carbon fibre reinforced epoxy matrix with 57.4% volumetric fibre ratio.

Figure 2. Representative volume elements used in this study

uz (TOP)-uz (BOTTOM)=0 (1) uy (RIGHT)-uy (LEFT)=0 (2) ux (FRONT)-ux (BACK)=0 (3)

Table 1. Boundary conditions to be applied in axial loading type

Table 2. Boundary conditions to be applied in shear loading type

2.2. Calculation of Elastic Coefficients Because composite materials are transversely isotropic materials, they have a total of five mechanical elasticity coefficients. These are listed in Table 3. Although a total of six parameters are listed in this table, the transverse shear modulus, G23, can be calculated using other coefficients with equation 4. To find the relevant elasticity coefficient, a unit value strain is applied to provide the necessary deformation, and as a result, the average value of the stress distribution created in the model equals the relevant elasticity modulus. Elasticity moduli are calculated by applying the following three steps: i. The "equation type constraint" sums the stresses on the plane as reaction force at a single node point. This force value should first be noted, ii. The reaction force at this single node point should be divided by the plane area to obtain the average stress value, iii. Then, by applying Hooke's law, the average stress value is divided by the applied strain value to calculate the relevant elasticity modulus. Since the applied strain value is a unit value (1), the average strain value calculated in (ii) is actually the desired elasticity modulus value.

Table 3. Elasticity coefficients of composite materials

3. Results and Discussion

The deformations and stress distributions resulting from the application of different types of unit value strains are shown in Figure 3. In this figure, only hexagonal arranged representative volume elements are shown for the model where fibres are reinforced in a single direction. This is because the transverse shear modulus value G23 obtained with the square arranged representative volume element does not give the same result with equation 4. For this reason, the stress distributions in the square arranged model may not be entirely correct. Although this article addresses only the calculation of elasticity moduli using the finite element micromechanics analysis method, the capability of this method is not limited to this alone. Residual stresses resulting from manufacturing effects and the progressive damage mechanism resulting from mechanical loads applied after these residual stresses, and their effects on strength, can also be simulated and predicted using this method. Detailed studies containing these topics exist in the literature [4–7]. Readers wishing to have more detailed information about these topics and models can contact the authors.

Figure 3. Deformations and stress distributions

Table 4. Comparison of elasticity coefficients with test results

4. Conclusion

This article summarizes the capabilities of the finite element micromechanics analysis method that makes predictions about the mechanical behaviour of materials at the micro-scale. A thermoset epoxy composite material with fibres reinforced unidirectionally and cross-ply has been investigated using square and hexagonal arranged RVEs. Elasticity moduli have been calculated using the finite element micromechanics analysis method very close to good experimental results. With the finite element micromechanics analysis method, stress distributions occurring in fibre and matrix under different loadings can be obtained at micro-scales, and insight into how and where damage will occur can be gained. This method can also be applied to investigate residual stresses resulting from manufacturing and their effects on progressive damage. Dr. Fatih Ertuğrul Öz Department of Mechanical Engineering Faculty of Engineering Boğaziçi University Project Leader Kordsa   Prof. Dr. Nuri Ersoy Department of Mechanical Engineering Faculty of Engineering Boğaziçi University Department of Design and Mathematics Faculty of Engineering University of the West of England  

5. References

1. Abaqus 6.14, "Documentation." Dassault Systèmes, 2014, 2014. 2. Xia, Z., Y. Zhang, and F. Ellyin, "A unified periodical boundary conditions for representative volume elements of composites and applications", International Journal of Solids and Structures, Vol. 40, No. 8 pp. 1907–1921, 2003. 3. Ersoy, N., T. Garstka, K. Potter, M. R. Wisnom, D. Porter, M. Clegg, and G. Stringer, "Development of the properties of a carbon fibre reinforced thermosetting composite through cure", Composites Part A: Applied Science and Manufacturing, Vol. 41, No. 3 pp. 401–409, 2010. 4. Oz, F. E., Micromechanical Progressive Damage Model for Predicting Resin Dominated Strength Values of Fibre Reinforced Composites Under Various Types of Loading, Master's Thesis, Boğaziçi University, 2012. 5. Ersoy, N. and F. E. Oz, "Micromechanical Investigation of Residual Stresses and Strength of Cross-Ply Laminates," ICCM 19 - Proceedings of the 19th International Conference on Composite Materials, Montreal - Canada, 2013. 6. Oz, F. E. and N. Ersoy, "Acoustic Emission Analysis for Validation of Micro Mechanical Models," ICCM 2015 - Proceedings of the 20th International Conference on Composite Materials, Copenhagen - Denmark, 2015. 7. Oz, F. E., Characterisation of Failure in Composite Materials with Acoustic Emission and Correlation with Micromechanics, Doctoral Thesis, Boğaziçi University, 2018. 8. Marlett, K., Y. Ng, and J. Tomblin, "Hexcel 8552 AS4 Unidirectional Prepreg at 190 gsm & 35 % RC Qualification Material Property Data Report", Niar - Wichita State University, 2011.